The Wanderers' win took them back to the top of the A-League, two points ahead of Brisbane Roar, which has a game in hand.
Heart's loss, in front of 8,070 fans that braved the rain swept conditions, leaves it bottom of the competition with two points from five games.
Wanderers coach Tony Popovic sprang a surprise by giving Matthew Spiranovic his first start of the season in lieu of captain Michael Beauchamp.
He also dropped striker Mark Bridge in favour of Cole.
The home team attacked furiously from the start and pegged the Wanderers in their half.
Golgol Mebrahtu got behind the Wanderers' defence in the 20th minute but his low cross found no takers.
At the other end Andrew Redmayne had to dive at Brendon Santalab's feet to avert a dangerous situation.
Heart's Michael Mifsud sprang into action soon after when he lost Spiranovic with a deft turn but the Malta striker could only hit the side netting from a strong position.
Heart was made to pay dearly for the miss barely a minute later when Cole, in his first start of the season, put the Wanderers ahead with an angled drive drive past Andrew Redmayne.
The Wanderers should have extended their lead on the half-hour when Spiranovic came from the blind side of Heart's defence and hit the bar with a header, but Santalab headed the ball over from a metre out.
David Williams was proving a thorn to the Wanderers defence with his pace and he was desperately unlucky not to score in the 39th minute when his dipping shot from 30 yards clipped the bar.
Four minutes into the second half Spiranovic was forced to deflect an awkward Williams cross to corner as Heart threatened to take the initiative.
Aaron Mooy reminded Heart of the Wanderers' potential to counter-attack when his firm drive shaded the post after a swift break.
Heart was forced to up the tempo to save the game and fullback Aziz Behich brought the best out of Ante Covic with a fierce shot 20 minutes from time.
With time running out, substitute Heart striker Mate Dugandzic hit the bar with a header.
Tomi Juric was denied the chance to seal the game by a Redmayne save.
"I'm happy we kept a clean sheet but I'm buggered to be honest," Spiranovic said.
"I only knew yesterday that I was playing so that came as a bit of a surprise."
Heart coach John Aloisi lamented another night of missed scoring chances.
"We are playing well but we need to punish teams," Aloisi said.
"We had a few injuries today but we need to work harder to improve our finishing."